A symbol, in this document, means a continuous time waveform with a fixed duration, called “symbol duration,” or a discrete time waveform with a fixed number of samples, called chips at the Nyquist sampling rate. It is assumed that there exists a finite set of symbols and that there exists a one-to-one and onto mapping from the transmitted information bits to the symbol set.
A received signal through a multi-path channel may be resolved into a number of replicas of the transmitted signal with different delays. The power-delay profile of a multi-path channel shows the number of replicas, their relative delays according to the earliest one, and their average powers.
An equivalent channel impulse response for a multi-path channel may be calculated accurately at the sampling rate in the receiver. In that case, the time interval between two channel response taps becomes a multiple of the reciprocal of the sampling rate. The tap corresponding to the highest amplitude is called the cursor. The taps prior to the cursor are called the pre-cursors and the taps after the cursor are called the post-cursors.
In general, the power-delay profile is supposed to be an exponentially decaying curve. That is, the pre-cursors disappear on average. However, the pre-cursor part can exist with a low probability. A pessimistic example is that the channel impulse response may be symmetric around the cursor and have a length of two symbol durations. The energy of the sth symbol may then be spread over the previous symbol and the next symbol. Hence, the observation interval in the detection of the sth symbol becomes the three-symbol interval comprising s−1, s, and s+1 in order to use all received energy belonging to the symbol s. This also implies the energies of the neighbor symbols are spread over the observation interval of a received symbol; hence, inter-symbol interference occurs in this situation. Hereinafter, the interference caused by previous symbols is called Previous Symbol Interference (PSI), and the interference caused by next symbols is called Next Symbol Interference (NSI).
Another case is the minimum phase channel impulse response, that is, the channel impulse response without the pre-cursor part. The sth symbol in this situation is spread over the interval of the S+1th symbol. Therefore, the optimal observation time becomes the two-symbol interval comprising s and s+1.
A detection method proposed in the literature is based on Maximum Likelihood (ML) (See J. G. Proakis, Digital Communications, 3rd ed., New York: McGraw-Hill, 1995). ML detection is reduced to the minimum distance problem if the additive noise in the received samples is independent zero-mean Gaussian, which is widely accepted.
For example, the modulator output may be a continuous stream of symbols x(s), which are selected from a finite symbol set. Linear channel experiences frequency selective distortion onto the transmitted signal; hence, the symbol energies are spread over time. One solution in the sense of ML uses a matched filter and a Viterbi algorithm with a search depth proportional to the expected maximum root mean square multi-path spread. The number of states in the Viterbi algorithm is equal to the number of elements in the symbol set. Thus, the Viterbi algorithm can become complex if the number of elements in the symbol set is high.
An ML detection method under multi-path (frequency selective fading) is the rake receiver if the symbol set satisfies the following orthogonality properties: (i) two different symbols in the set are orthogonal to any delayed version of themselves; (ii) a symbol in the set is orthogonal to any non-zero delayed version of itself; (iii) the symbols in the symbol set have identical energy; and (iv) the symbol duration is long relative to multi-path spread. If these properties fail in some degree, then the rake receiver cannot completely cancel inter-symbol interference (ISI) and the correlator bank does not provide equivalent metrics for Maximum Likelihood detection due to inter-chip interference (ICI). Thus, the rake receiver cannot provide path diversity in an ideal manner. The imperfection of the orthogonality properties (i) and (ii) is explained as a self-noise (See J. G. Proakis, Digital Communications, 3rd ed., New York: McGraw-Hill, 1995). If all four of the orthogonality properties are satisfied, then the rake receiver can resolve the multi-path and provide the path diversity. However, if N is small, then the imperfection in the orthogonality properties is unavoidable.
U.S. Pat. No. 6,233,273 claims an improved rake receiver structure with an Embedded Decision feedback Equalizer (DFE) for direct sequence spread spectrum (DSSS) transmissions. A DFE is employed to remove inter-symbol interference with or without feed-forward taps located between the channel-matched filter and the correlator bank. The channel-matched filter and the feed-forward taps can be convolved to be a single filter. The term “canceling inter-chip interference” is used in the patent along with “canceling inter-symbol interference.” However, the cancellation of inter-chip interference conflicts with the path diversity achieved by the rake receiver for DSSS transmissions. Moreover, although the feedback taps can be designed to cancel inter-symbol interference only, this is not completely valid for the feed-forward taps. The filtering through feed-forward tries to cancel inter-chip interference, which again conflicts with the path diversity that is a goal of the design of rake receiver.